Questions tagged [mgf]

The moment generating function (mgf) is a real function which allows to derive a random variable's moments and therefore can characterize its entire distribution. Use also for its logarithm, the cumulant generating function.

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49 views

Gamma Distribution satisfying property

How can we prove that gamma random variable $X_{n}$ with parameters $(n,3)$ can satisfy the following relation for some $n$? $$P(X_{n} < n/2) > 0.999$$ I used the definition of density function ...
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Expectation of the product of polynomial & exponential transformations of normal r.v

Let $X \sim \mathcal{N}(\mu, \sigma^2)$. Are there any (1) general formula and (2) references to the general formula for $$ \mathbb{E} (X^n e^{tX}),\; n \in \mathbb{N}, t \in \mathbb{R}$$ in ...
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37 views

The log of a Moment Generating Function

I am having trouble understanding the rationale of the following related to the MGF: Function Mx(t)=E[exp(tX)], the expectation exists for all t in a neighbourhood of zero, and X has mean mx, show ...
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How would a moment generating function change if all the random variables are increased by a value

Suppose you have some moment generating function $M_x(t)$ Now all the random variables x are increased by a arbitrary value b. What is the new moment generating value? I tried solving this by moving ...
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1answer
48 views

Weak convergence of moment generating function

I have the following sequence of rvs $$Z_1 = X_0*Y_0$$ $$Z_{n+1} = Z_n /2 + X_n*Y_n$$ Where $X_n$ and $Y_n$ are independent, with $X_n$ having Bernoulli distribution with p=1/2 and $Y_n$ having ...
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33 views

Inverting a moment generating function

If I have a random variable $X$ with pdf $f$ I can compute its MGF as $$ M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)dx. $$ My understanding is that this is basically a Laplace transformation. ...
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1answer
25 views

Sum of indepedent random variables and a constant

Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational ...
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How to prove there is no moment generating function for t distribution

I'm struggling to show that the moment generating function for t distribution does not exist. So far I tried to show the moment generating function diverges from its integration but the computation is ...
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32 views

How to calculate mean and covariance of bivariate normal?

The moment generating function is $$ M_{X_1X_2}(t_1,t_2)= e^{t^T\mu+0.5t^T\Sigma t}$$ where $t^T = \pmatrix{t_1\\t_2}$. How to calculate mean and covariance now? I have the mean... $$ \frac{\partial}{\...
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1answer
62 views

How do you calculate the expected value of $E\left\{e^{-|X|}\right\}$ e.g. for Gaussian X?

If $X$ is a random variable, I would like to be able to calculate something like $$E\left\{e^{-|X|}\right\}$$ How can I do this, e.g., for a normally distributed $X$?
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Example of random variable with a unique moment sequence but mgf DNE in a neighborhood of 0

Do you have an example of a random variable $X$ with a unique moment sequence but whose mgf does not exist in a neighborhood of 0? In other words, I'm looking for a counterexample to the converse of ...
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121 views

Moment generating function of a conditional distribution

Let $S$ ~ Poisson$(\alpha + \beta)$, and $X|_{S = s}$ ~ Binomial$(s, \alpha/(\alpha + \beta))$, $\alpha > 0, \beta > 0$ Suppose Z = S - X is independent from X. What is the distribution of Z? ...
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1answer
32 views

n'th cumulant (of a CGF) for exponential family / exponential dispersion model

The n'th cumulant is defined to be the n'th derivative of the CGF (cumulant generating function). $$\kappa_n = \frac{d^n K(t)}{dt^n} |_{t=0} $$ But I'm reading in a book (p.215, chapter5, eq. 5.8) ...
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Can we get Moment Generating Function(MGF) from data?

We had couple of good discussions about Moment Generating Function(MGF), here and here. But I still have questions on the applications of it and how can it be useful. Specifically, I can understand ...
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122 views

Tail bounds for sample means of i.i.d random variables where the moment generating function exists

I want to figure out the proof of Lemma 4 in the following paper. The lemma states that Let $Y_1, Y_2,\ldots Y_m$ be i.i.d random variables such that $\mathbb{E}\left[e^{zY}\right] < \infty$ ...
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1answer
57 views

Undefined MGF but all moments finite?

For the lognormal distribution: https://en.wikipedia.org/wiki/Log-normal_distribution The moment generating function is undefined, but all the moments exist and are finite. I thought the moment ...
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1answer
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How to show that $\frac{X_n-a}{\sqrt n}$ approaches a Gaussian (if $X_n \sim\ \chi_{n-p}^2$)

Given that $X_n \sim\ \chi_{n-p}^2$, I think the moment generating function of $\frac{X_n-a}{\sqrt n}$ is $e^{\frac{-at}{\sqrt n}}(1-\frac{2t}{\sqrt n})^{-\frac{n-p}{2}}$. As specified in problem 4 on ...
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Finding the MGF of a bivariate Normal Distribution [duplicate]

Given ($X$, $Y$) whose MGF is defined as: $$M_{XY}(s, t)=E[e^{sX+tY}]$$ Find $M_{XY}(s, t)$ when $X$ and $Y$ are two jointly normal random variables with $E[X]=\mu_X$, $E[Y]=\mu_Y$, $var(X)=\sigma^...
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1answer
159 views

Distributions of Quadratic form of a normal random variable

I am looking for ways to prove that the moment generating function of $X'AX$ given that $X \sim N(\vec{\mu}, \vec{\Sigma})$ and $A$ is symmetric is defined as: $$M_{X'AX}(\vec{t})= \frac{1}{|I-2tA\...
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Generate Moments of Continuous Uniform Distribution with Moment Generating Functions

I am having trouble generating moments from the moment generating function of the uniform. By the definition of M.G.F, we can calculate: $$ M(t) = \begin{cases} \frac{e^{tb} - e^{ta}}{tb-ta} : t \ne ...
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Generating function of relaxed Bernoulli

In [1,2], an approximate continuous relaxation of the Bernoulli distribution is introduced as follows: $$X = \frac{1}{1 + e^{- (\theta - L) / \tau}}$$ where $L$ is a Logistic random variable, $\tau&...
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Moment generating function of non-central Chi-squared distribution with complex mean?

I have random variables $(X_1, \dots, X_k)$ distributed independently according to normal distributions with complex means, i.e. $j\mu_i, i=1\dots k, j^2=-1$, with unit variances. I want to study the ...
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207 views

How to find the MGF of the difference of 2 random variables

Let $X\sim N(12,4)$ and $Y \sim N(3,1)$ Let $Z = X - Y$ Find the Moment Generating Function of $Z$. I tried finding the expected value of $e$ to the power of $tz$, but this isn't possible to separate ...
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Why is moment generating function (MGF) evaluated at zero?

Although this question touched on MGF exists at neighborhood of 0, I still don't understand why the definition of MGF says $M_X{(t)}$ = E $e^{tX}$ for $t$ in neighborhood of 0. Why must it be 0 but ...
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1answer
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Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...
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1answer
65 views

Distribution of sum of independent random variables using MGF

Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$. and let $Z=X_1+ \dots +X_n$ and I wanted to show that $Z$, $Z \sim \operatorname{...
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Self-Study: If $x \sim \mbox N_p(\mu; V)$ and A is a matrix then how to show that $E[(x-\mu)(x-\mu)'A(x-\mu) = 0$?

If we assume the random vector $x$ to be normally distributed with $N_p(\mu; V)$ then $E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p$. If I am not mistaken, this can be shown using the moment generating function of ...
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Getting the pdf from a Moment generating function

I have to solve the following question: Let $X$ follow the distribution with moment generating function $M_X(t)$ and Let $Y = aX + b$ follow the distribution with moment generating function $...
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1answer
65 views

Conditioning on MGF

Suppose $Z_i$ is the total loss from all losses on policy $i$, where $q_i=P(there\ are\ losses\ from\ policy\ i),\ i=1, \dots, n.$ Then $X_i$, the total loss on policy $i$ can be defined as $...
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1answer
71 views

Obtain the moment generating function

How do I come up with the mgf given this: $E(X^r)=\frac{(r+1)!}{2^r} , r = 1, 2, 3, ...$ ? The exercise does not require to prove that the distribution exists. It only asks to obtain the mgf and its ...
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162 views

Using the law of total expectation and the definition of the MGF to find the unconditional distribution

During my research, I encountered Jarle Tufto's answer in this question on the MGF of conditioned random variables: The mgf of $Y$ conditional on $N=n$ is $$ M_{Y|N=n}(t)=M_X(t)^n, $$ since ...
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How do I find all even moments (and odd moments) for $f_X(x)=\frac{1}{2}e^{-|x|}$?

I was asked to find a formula for all even moments of the form $E(X^{2n})$ and all odd moments of the form $E(X^{2n+1})$ using a mgf. Can you help me find the even moments? I will attempt to solve for ...
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1answer
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Where does the constant “B” come from in moment generating fuctions?

In my book "Mathematical Statistics with Applications" by Dennis Wackerly it's stated that the moment generating function exists if The moment-generating function m(t) for a random variable Y is ...
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55 views

Exponential MGF

Let $T$ be a random variable with the probability density function (PDF) $f(t) = \lambda e^{−λ(t−a)}$, $\lambda > 0$, $t > a$. Find the moment generating function (MGF) of $T$. This appears to ...
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1answer
135 views

Moment Generating Function

Suppose X is a random variable with a Beta distribution and x in (0,1) How can I prove moment generating function exist
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Find mgf from joint pmf

The joint pmf of random variables $ X$ and $ Y$ is given by $$p_{XY}(x,y)= \begin{align} & \frac{e^{-2}}{x! (y-x)!}\quad\text{if}\,\,\,x= 0,1,...y,\ y=0,1,... \\ \end{align} $$ Find its mgf. \...
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1answer
72 views

Finding the marginal distribution

If $p(y) =N(0,\Gamma) $ and if $p(x|y) =N(\Lambda y, \psi I) $where $I$ denotes the identity matrix. Also, we have the conditions: $\Gamma_{ij} =0$ for $i \neq j$ (diagonal) $\Lambda \Lambda^T = I$ ...
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Why can we treat MGF in this way

For the standard proof that if $Z \sim N(0,1)$ than $Z^{2} \sim \chi^{2}_{1}$ We write: $$M_{Z^{2}}(t)=\int_{\mathbb{R}}\exp(tz^{2})\frac{1}{\sqrt{2\pi}}\exp(\frac{-z^{2}}{2}) dz$$ That is, we use ...
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Moment Generating Function for Lognormal Random Variable

I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. I understand the following: Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our ...
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Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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1answer
741 views

Relations between moments and cumulants

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= ...
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1answer
53 views

Moment generating function of binomial distribution

I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomial distribution iwth $p=\frac{1}{2}$. With the standardized test statitics is $S=\frac{S(\theta_0)-(\frac{n}{2})}{...
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1answer
602 views

Distribution of $\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$ where $X_i,Y_i$s are i.i.d Normal variables

Suppose $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ are i.i.d $\mathcal N(0,1)$ random variables. I am interested in the distribution of $$U=\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$$ I define $$Z=\frac{\...
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1answer
122 views

Moment generating function between 2 variables

I understand that mean of Y is M'Y(0), whereas variance of Y is M''Y(0). I can derive expressions through differentiation to get M'Y(0) and M''Y(0). The 2, however, have the expression Mx(0) in them. ...
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632 views

How does MGF of Pareto distribution of first kind exist for non-positive values of t? [closed]

I have reached upto the stage shown in the attached picture. The r.v. X is always positive and its power $\beta+1$ is also always positive. Therefore, how can it be said that MGF exists for t <= 0? ...
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4answers
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In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
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1answer
55 views

Sum of negative binomial variables with transformed parameters

Let $X_i:i=1,2,...,n$ be independent ~ $NegBin(\mu,\alpha)$random variables such that $E(X_i) = \mu$, $Var(X_i) = \mu + \frac{\mu^2}{\alpha}$. (i) Find the mean and variance of $Y=∑(X_i)$. (ii) Find ...
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1answer
186 views

Tail bound for sum of i.i.d. random variables with common moment generating function

Suppose $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of independent and identically distributed random variables and $S_n:=X_1+...+X_n$. Assume that each $X_i$ has mean $0$ and that all $X_i$ have a ...

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